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					Bose-Einstein condensates
in optical lattices
and speckle potentials
Michele Modugno
Lens & Dipartimento di Matematica Applicata, Florence
CNR-INFM BEC Center, Trento

BEC Meeting, 2-3 May 2006
Part I: Effect of the transverse confinement
on the dynamics of BECs in 1D optical lattices

A) Energetic/dynamical instability
M. Modugno, C. Tozzo, and F. Dalfovo, Phys. Rev. A 70, 043625 (2004);
Phys. Rev. A 71, 019904(E) (2005).

L. Fallani, L. De Sarlo, J. E. Lye, M. Modugno, R. Saers, C. Fort, and M. Inguscio,
Phys. Rev. Lett. 93, 140406 (2004).

L. De Sarlo, L. Fallani, C. Fort, J. E. Lye, M. Modugno, R. Saers, and M. Inguscio,
Phys. Rev. A 72, 013603 (2005).

B) Sound propagation
M. Kraemer, C. Menotti, and M. Modugno, J. Low Temp. Phys 138, 729 (2005).

  • Theory: 1D models
     – 1D GPE: energetic/dynamical instability [Wu&Niu, Pethick et al.],
               Bogoliubov excitations, sound propagation [Krämer et al.]
     – DNLSE (tight binding): modulational (dynamical) instability [Smerzi et al.]

  • Experiment: Burger et al. [PRL 86,4447 (2001)]:
     – breakdown of superfluidity under dipolar oscillations interpreted
       as Landau (energetic) instability

  • Effect of the transverse confinement ?
     – Need for a framework for quantitative comparison with experiments
       both in weak anf tight binding regimes
     – Clear indentification of dynamical vs energetic instabilities
     – Role of dimensionality on the dynamics (3D vs 1D)
         Energetic (Landau) vs dynamical instability
 Stationary solution + fluctuations:           Time dependent fluctuations:

                                                Linearized GPE -> Bogoliubov equations:

  Negative eigenvalues of M(p) -> (Landau)     Imaginary eigenvalues -> modes that
 instability (takes place in the presence of   grow exponentially with time
 dissipation, not accounted by GPE)
A cylindrical condensate in a 1D lattice

  3D Gross-Pitaevskii eq.

                                        harmonic confinement + lattice

 -> Bloch description in terms of periodic functions

   Bogliubov equations -> excitation spectrum
p=0: excitation spectrum, sound velocity

                                         Radial breathing
                                         Axial phonons

 Excitation spectrum (s=5): the lowest        Bogoliubov sound velocity of the lowest
 two Bloch bands, 20 radial branches          phononic branch vs the analytic prediction
Velocity of sound from a 1D effective model

  •Factorization ansatz:                   -> two effective 1D GP eqs:

                                                                 axial -> m*, g*

                                                                 radial -> µ(n)


                                                          QuickTime™ an d a
                                                      TIFF (LZW) decompressor
                                                   are need ed to see this p icture .

   Exact in the 1D meanfield (a*n1D <<1)
   and TF limits (a*n1D >>1)
                                           GPE vs 1D effective model
                                           (s=0,5,10 from top to bottom)
P≠0: excitation spectrum, instabilities

                                         Phonon-antiphon resonance = a conjugate
                                         pair of complex frequencies appears
                                         -> resonance condition for two particles
                                         decaying into two different Bloch states
  Real part of the excitation spectrum
                                         E1(p+q) and E1(p-q) (non int. limit)
  for p=0,0.25,0.5,0.55,0.75,1 (qB)
NPSE: a 1D effective model

 3D->1D: factorization + z-dependent Gaussian ansatz for the radial component

 -> change in the functional form of nonlinearity
 (works better that a simple renormalization of g)

 Effect of the transverse trapping through a
 residual axial-to-radial coupling

  Same features of the =0 branch of GPE
Stability diagrams


                                     energetic instab.
 Excitation quasimomentum
                                     en. + dyn. instab.

                                 Max growth rate
             BEC quasimomentum
Revisiting the Burger et al. experiment
 Dipole oscillations of an elongated BEC in magnetic trap + optical lattice (s=1.6)
    – lattice spacing << axial size of the condensate ~ infinite cylinder
    – small amplitude oscillations: well-defined quasimomentum states
-> Quantitative analisys of the unstable regimes    + 3D dynamical simulations (GPE)

  Center-of-mass velocity vs BEC quasimomentum.        Center-of-mass velocity vs time.
  Dashed line: experimental critical velocity          Density distribution as in experiments
                                                       (in 1D the disruption is more dramatic)
-> Breakdown of superfluidity (in the experiment)
  driven by dynamical instability
BECs in a moving lattice
 By adiabatically raising a moving lattice -> project the BEC on a selected Bloch state
 -> explore dynamically unstable states not accessibile by dipole motion

                                                        S=0.2                    S=1.15
 The (theoretical) growth rates show
 a peculiar behavior as a function of
 the band index and lattice heigth
 Similar shapes are found in the loss
 rates measured in the experiment
 -> the most unstable mode imprints
 the dynamics well beyond the linear regime
Beyond linear stability analysis: GPE dynamics

                                        Density distribution after expansion:
                                        theory (top) vs experiment @LENS
                                        -> momentum peaks hidden in the

                                        Recently observed at MIT
                                        (G. Campbell et al.)
 Growth and (nonlinear) mixing of the
 dynamically unstable modes
Conclusions & perspectives

  Effects of radial confinement on the dynamics of BECs:
     Proved the validity of a 1D approch for sound velocity
     Dynamical vs Energetic instability
     3D GPE + linear stability analysis: framework for quantitave
      comparison with experiments
     Description of past and recent experiments @ LENS

 • Attractive condensates: dynamically unstable at p=0, can be
   stabilized for p>0?

 • Periodic vs random lattices……
Part II:
BECs in random (speckle) potentials

M. Modugno, Phys. Rev. A 73 013606 (2006).

J. E. Lye, L. Fallani, M. Modugno, D. Wiersma, C. Fort, and M. Inguscio, Phys.
Rev. Lett. 95, 070401 (2005).

C. Fort, L. Fallani, V. Guarrera, J. E. Lye, M. Modugno, D. S. Wiersma, and M.
Inguscio, Phys. Rev. Lett. 95, 170410 (2005).

 •   Disordered systems: rich and interesting phenomenology
     – Anderson localization (by interference)
     – Bose glass phase (from the interplay of interactions and disorder)

 • BECs as versatile tools to revisit condensed matter physics
   -> promising tools to engineer disordered quantum systems

 • Recent experiments with BECs + speckles
    – Effects on quadrupole and dipole modes
    – localization phenomena during the expansion in a 1D waveguide

 • Effects of disorder for BECs in microtraps
A BEC in the speckle potential
BEC radial size < correlation length (10 µm) -> speckles ≈ 1D random potential

                                                                 intensity distribution
                                                                 ~ exp(-I/<I>)

                 A typical BEC ground state in the harmonic+speckle potential
Dipole and quadrupole modes
Sum rules approach, the speckles potential as a small perturbation:

                                                            -> uncorrelated shifts

 Dipole and quadrupole frequency shifts for 100    random vs periodic: correlated shifts (top),
 different realizations of the speckle potential   but uncorrelated frequencies (bottom) that
                                                   depend on the position of the condensate in
                                                   the potential.
GPE dynamics

Dipole oscillations in the speckle potential (V0=2.5 —wz):

             Sum rules vs GPE

Small amplitudes: coherent undamped oscillations.
Large amplitudes: the motion is damped and a breakdown of superfluidity occur.
Expansion in a 1D waveguide
red-detuned speckles vs periodic:
• almost free expansion of the wings (the
most energetic atoms pass over the
• the central part (atoms with nearly
vanishing velocity) is localized in the
initially occupied wells
• intermediate region: acceleration across
the potential wells during the expansion
•The same picture holds even in case of a
single well.

blue-detuned speckles (Aspect experiments):
• reflection from the highest barriers that eventually stop the expansion
• the central part gets localized, being trapped between high barriers

-> localization as a classical effect due to the actual shape of the
Quantum behavior of a single defect

Single defect ~                            -> analytic solution (Landau&Lifschitz)

Incident wavepacket of momentum k:
quantum behaviour signalled by 2|0.5-T(k, ab

(a)-(b): potential well, (c)-(d): barrier
(a)-(c) a=0.2, (b)-(d) a=1.
Dark regions indicate complete reflection or
transmission, yellow corresponds to a 50%

Current experiments (ß~1) : quantum effects only in a very narrow range close to the top
of the barrier or at the well border. By reducing the length scale of the disorder by an
order of magnitude (ß~0.1) quantum effects may eventually become predominant.
Conclusions & perspectives

 • BECs in a shallow speckle potentials:
    – Uncorrelated shifts of dipole and quadrupole frequencies
    – Classical localization effects in 1D expansion
    (no quantum reflection)

   ->reduce the correlation length in order to observe
     Anderson-like localization effects
   -> two-colored (quasi)random lattices

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